Groupoids and C*-algebras for categories of paths

TL;DR

This paper introduces a novel method for constructing C*-algebras from combinatorial path data, extending existing frameworks for graphs and higher-rank structures, and provides conditions for their nuclearity and uniqueness.

Contribution

It generalizes C*-algebra constructions from directed graphs to categories of paths using elementary path operations, with new criteria for AF cores and aperiodicity.

Findings

Established a new general framework for C*-algebras from path categories.

Provided sufficient conditions for nuclearity and AF cores.

Proved standard uniqueness theorems under aperiodicity.

Abstract

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.